11.2.1Dynamics

Consider a satellite as a mass point moving around the Earth, its state in a geocentric inertial cartesian coordinate (GICC), illustrated in Figure 1, consists of its position vector r ∈ ℝ3{0}, velocity vector υ ∈ ℝ3, and mass Then, the dynamics for themotion of the satellite for positive times can be written as

where μ > 0 is the gravitational constant, β > 0 is a scalar constant determined by the specific impulse of the low-thrust control system equipped on the satellite, ‖ ⋅ ‖ denotes the Euclidean norm, and the thrust (or control) vector τ ∈ ℝ3 takes values in the admissible set

where τmax is a positive constant.

If ? = ℝ3{0} × ℝ3 and x = (r, υ), we define two vector fields f 0, f 1 on ? by

with ℝ6×3 denotes the set of 6 ×3 matriceswih real entries and I 3 is the identity matrix of ℝ3. Let ℬε be the closed ball in ℝ3 centered at the origin and of radius ε > 0. For every ε > 0, we consider the control-affine system Σε given by

where the control vector u ∈ ℝ3 takes values in ℬε.Wewill use in this paper the vector field point of view of [11, 13, 14]. For every point x ∈ ? and every u ∈ ℬε, we denote by

where f 0 and f 1 are referred to as the drift vector field and the control vector field, respectively. Note that trajectories of Σε starting at any x0 ∈ ? and associated with a measurable u : ℝ+ → ℬε are well defined on an open interval of ℝ+ containing 0, which depends in general on x0 and u(⋅).

11.2.2Study of the drift vector field in X

In this section, we recall the main properties of the drift vector field f 0. For every x ∈ ?, we use γx to denote the restriction to ℝ+ of the maximal trajectory of f 0 starting at x, that is, γx is defined on some interval [0, tf (x)), where tf (x) ≤∞. Then, the following holds true.

Property 2.1 (First integrals [2, 6]). For every x ∈ ?, if γx(t) = (r̃(t), ῡ̃(t)) on [0, tf (x)), the quantities

are constant along γx and the corresponding constant values are the angular momentum vector h ∈ ℝ3, the Laplace vector L ∈ ℝ3, and the mechanical energy of a unit mass E ∈ ℝ, which is the sum of the relative kinetic energy ‖ ῡ̃(t) ‖2 /2 and the potential energy −μ/ ‖ r̃(t) ‖.

Combining equations (2.7) and (2.8), we have the following two properties:

Property 2.2 (Straight line [2, 6]). Let x ∈ ? with h = 0, that is, r and υ are collinear. Then, the trajectory γx is a straight line and tf (x) is either finite or infinite depending on initial conditions.

Property 2.3 (Conic section [2, 6]). Let x ∈ ? with h ≠ 0, that is, r and υ are not collinear. Then, tf (x) is infinite and the trajectory γx is a periodic trajectory with locus defining a conic section lying in a two-dimensional plane perpendicular to h called the orbital plane.

Let

Then define on ? the function e : x ↦ ‖ L ‖ /μ. Along every trajectory of f 0 starting at x ∈ ?̃ , one gets, after multiplying equation (2.8) by r̃(t), that

where the angle θ(t, x) is defined by Note that the previous formula holds true if L = 0, since in that case, e(x) cos θ(t, x) is equal to zero and the orbit is a circle.

Notice from equation (2.11) that an orbit (r̃(t), ῦ̃(t)) = γx(t), with x ∈ ?̃ on ℝ+ being a parabola if e(x) = 1 and a hyperbola if e(x) > 1. Put a satellite on a parabolic or ahyperbolic orbit without any control; then, it can escape to infinity: Thus, parabolic and hyperbolic orbits are generally used for a satellite to escape from the gravitational attraction of the Earth. For every point x ∈ ?̃ , if 0 ≤ e(x) < 1, the orbit γx(t) on ℝ+ is an ellipse, whose orientation is illustrated in Figure 2. Moreover, it is easy to deduce the following characterization of elliptic orbits.

Property 2.4. Given every point x ∈ ?̃ , the mechanical energy E is negative if and only if e(x) < 1.

Thus, let us define the set

then for every point (r, υ) ∈ ?, the associated orbit γx on ℝ+ is periodic and the set ? is called the periodic region in ?.

Definition 2.5 (Smallest period tp). Given every point x ∈ ?, we denote by

tp : ?→ℝ, x → tp(x) ,

the smallest period of the orbit γx on ℝ+.

According to equation (2.11), for every point x ∈ ?, if e(x) ≠ 0, the associated orbit γx on [0, tp(x)] has its perigee point and apogee point at θ(t, x) = 0 and π, respectively. Thus, let

where rp(x) and ra(x) are the perigee and apogee distances of the orbit γx on [0, tp(x)], respectively, if e(x) ≠ 0. Note that ra(x) = rp(x) if and only if e(x) = 0, which corresponds to a circular orbit.

Property 2.6 (Minimum radius and maximum radius). Given every periodic orbit (r̃(t), ῦ̃(t)) = γx(t) on [0, tp(x)] in ?, we have rp(x) ≤‖ r̃(t) ‖≤ ra(x) on [0, tp(x)]. Thus, the perigee distance rp(x) and apogee distance ra(x) are the minimum radius and maximum radius of the orbit (r̃(t), ῦ̃(t)) on [0, tp(x)].

11.2.3Admissible controlled trajectory of Σsat

For every initial point and measurable control function τ(⋅) taking values in ?(τmax) with τmax > 0, we use Γ(t, τ(t), yi) to denote the restriction to ℝ+ of the solution of Σsat starting from yi. Let tf̃ ∈ ℝ+ be the maximum time such that the solution Γ(t, τ, yi) lies in and set ?Γ = [0, tf̃ ). Then the restriction of Γ(t, τ, yi) to ?Γ is said to be the controlled trajectory of Σsat starting from yi and associated with τ(⋅).

Remark 2.7. For every point , let (x(t), m(t)) = Γ(t, 0, yi) on ?Γ ; then, we have x(t) = γxi (t) and m(t) = mi for every t ≥ 0, that is, tf̃ = ∞.

Definition 2.8 (Controlled Keplerian motion). Given every initial point yi = (xi , mi) ∈ ?×ℝandmeasurable control function τ(⋅) taking values in ?(τmax) with τmax > 0, the corresponding trajectory Γ(t, τ(⋅), yi) of Σsat is called a controlled Keplerian motion.

Let rc > 0 and M0 > 0 denote the radius of the surface of atmosphere around the Earth and the mass of a satellite without any fuel, respectively; then, given every point on the trajectories of Keplerian motions, it is required that ‖ r ‖> rc and m > M 0.

Definition 2.9 (Admissible region). We define the set

the admissible region in ? for Keplerian motion and/ or controlled Keplerian motion.

Definition 2.10 (Admissible controlled trajectory). Given every M0 > 0, we say thatthe controlled trajectory (x(t), m(t)) = Γ(t, τ, xi , mi) of Σsat on some finite intervals[0, tf] ⊂ ?Γ with the initial condition is an admissible controlledtrajectory if (x(t)) ∈ ? and m(t) ≥ M0 for t ∈ [0, tf ].

For every time interval [0, tf ]⊂ ?Γ , since ṁ (t) ≤ 0, it follows m(tf )≤ m(t) on [0, tf ]. Thus, the inequality m(t) ≥ M0 can be ensured by m(tf) ≥ M0.

11.2.4Controlled problems in A

For x ∈ ?, let (r̃(t), ῦ̃(t)) = γx(t) on ℝ+; then, we have that the inequality ‖ r̃(t) ‖> rc is satisfied on ℝ+ if rp(x) > rc. Thus, we define the set

It is immediate to see that the periodic uncontrolled trajectory γ(t, x) starting at any x ∈ ?+ remains in ?+.

Let

Then, for every point x ∈ ?−, there exists an interval [t1, t2] ∈ [0, tp(x)] such that ‖ r̃(t) ‖≤ rc for t ∈ [t1, t2]. Thus, placing a satellite on a point x ∈ ?−, it can move out of the admissible region ?.

Definition 2.11 (Stable periodic region ?+ and unstable periodic region ?− in ?). Wesay that the two sets ?+ and ?− are the stable and unstable periodic regions, respectively.

All the satellites periodically moving around the Earth are located in the stable periodic region ?+. In order to fulfill observation or other mission requirements, a satellite is controlled to move from one point xi in ?+ to another point xf in ?+ by its control system.

Definition 2.12 (Orbital transfer problem (OTP)). We say that the problem of controlling a satellite from a point xi in ?+ to another point xf in ?+ is the OTP; see Figure 3a.

For a typical space mission, in order to place a satellite into a stable orbit in ?+, a rocket is used to carry the satellite from the surface of the Earth to a point xi in ?−, at which the rocket and the satellite are separated. From this moment on, the satellite is controlled by its own control system to be inserted into a stable orbit in ?+.

Definition 2.13 (Orbital insertion problem (OIP)). We say that the problem of controlling a satellite from an initial point xi ∈ ?− to a final point xf ∈ ?+ is the orbit insertion problem; see Figure 3b.

After a satellite in the stable region ?+ finishes its mission, it should be decelerated to return to the unstable region ?−. Then, the satellite will coast into atmosphere such that the aerodynamic pressure will act as a control to control the satellite to fly to landing sites.

Definition 2.14 (De-orbit problem (DOP)). We say that the problem of controlling a satellite from an initial point xi ∈ ?+ to a final point xf ∈ ?− is the de-orbit problem; see Figure 3c.

11.3.1Controllability for OTP

In this subsection, we first give a controllability property of Σε for OTP; then, according to Lemma 3.4, we will establish the controllability of Σsat for OTP.

Definition 3.7. For every controlled trajectory x̄(⋅) = Γε(⋅, ū , xi) of Σε (where ε > 0, ū(⋅) : [0, tf ]→ ℬε is measurable and xi ∈ ?), we define

(x̄) : λ̇(t) = A(t)λ(t) + B(t)u(t) ,

the linearized system along x̄(⋅) of Σε on [0, tf ], where

on [0, tf ].

We first have a result of local controllability for the systems Σε’s around the periodic trajectories of the drift vector field.

Lemma 3.8. Let x̄ ∈ ?+. Then, for every ρ > 0, there exists σ > 0 such that the following properties hold: ℬσ(x̄) ⊂ ?+ and, for every x ∈ ℬσ(x̄), there exists a controlled trajectory Γε(t, u(t), x̄) of Σε such that

and

for t ∈ [0, tp(x̄)].

Proof. According to Theorem 7 of Chapter 3 in [12], it suffices to prove the controllability of the linearized system along the periodic trajectory Γε(t, 0, x̄) on the interval [0, tp(x̄)]. Then, the latter controllability would follow. According to Corollary 3.5.18 of Chapter 3 in [12], by the following rank condition, there exists a time τ ∈ [0, tp(x̄)] and a nonnegative integer k such that the rank of the matrix [B0(τ), B1(τ), . . . , Bk(τ)] equals 6, where for i = 1, 2, . . ., and B0(t) = B(t). It, therefore, amounts to compute some Bi(⋅)’s. The explicit expressions for matrices A and B in terms of x are

Since B0(t) = B(t), it follows that

Thus, we have that the rank of the matrix equal to 6 forevery time t ∈ [0, tp(x̄)], proving the lemma.

Proposition 3.9. For ε > 0, the control system Σε is controllable for OTP within ?+, that is, for every initial point xi ∈ ?+ and final point xf ∈ ?+, there exists a controlled trajectory Γε(t, u, xi) of Σε in ?+ on a finite interval [0, tf ] ⊂ ?̄ such that Γε(tf , u, xi) =xf .

Proof. Since the subset ?+ is path-connected as is shown by Proposition 3.6, it followsthat any two different points xi and xf in ?+ are connected by a path x : [0, 1] → ?+, λ ↦ x(λ) such that x(0) = xi and x(1) = xf . By compactness of the support of x(⋅) in ?+, there exists, for every σ > 0, a finite sequence of points x0, x1, . . . , xN, on the support of x(λ) so that x0 = xi, xn = xf and xj+1 ∈ ℬσ(xj), for j = 0, 1, . . . , N − 1. According to Lemma 3.8, for every ρ > 0, there exists σ > 0 small enough and a finite sequence of points x0, x1, . . . , xN as above such that, for j = 0, 1, . . . , N − 1, xj+1 ∈ ℬσ(xj) and one has a controlled trajectory Γε(t, uj , xj) on the interval [0, tp(xj)] such that

and

For σ > 0, let ?j ⊂ ?+ be an open neighborhood of xj such that ℬσ(xj) ⊂ ?j forj = 0,1, . . . , N and set = {Γε(t, 0,?j ), t ≥ 0}. For ρ > 0 small enough, the openset is included in ?+. By concatenating the Γε(⋅, uj , xj), for j = 0,1, . . . , N − 1,the initial point xi can be steered to xf , proving the proposition.

According to Lemma 3.4, and recalling the definition of controllability for OTPs in Definition 2.12, we obtain the following result of controllability.

Corollary 3.10. For every μ > 0, β > 0, τmax > 0, the system Σsat is controllable for OTP.

Since ?+ ⊂ ?, the system Σsat is controllable for OTPs within ? for every positive τmax if the satellite takes high enough percent of total fuel, that is (mi − mf )/mi > 0 is big enough. This result makes senses in engineering for electric thrust systems whose maximum thrust τmax is very small.

11.3.2Controllability for OIP

We provide next a controllability criterion for OIP.

Lemma 3.11. Assume that, for every point there exists τ > 0 and a positive time t ̄ ∈ ?Γ , and a control τ̃(⋅) : [0, t]̄ → ?(τ) such that along the controlled trajectory (x̃, m̃ (t)) = Γ(t, τ̃(t), xi , mi) on [0, t]̄ , we have x̃(t) ∈ ? on [0, t]̄ , m̃ (t)̄ > 0, and rp(x̃(t)̄ ) > rc. Then, the system Σsat is controllable for OIP from (xi , mi).Proof. Note that the assumption implies that there exists a control τ̃(⋅) : [0, t]̄ → ?(τ) such that the admissible controlled trajectory (x̃(t), m̃ (t)) = Γ(t, τ, xi , mi) in ? × [m( t, ̄ mi] on [0, t] ̄ steers (xi , mi) in to some (x( t)̄ , m( t)) ̄ in After arriving at x(t)̄ in ?+, according to Proposition 3.9, it follows that there exists an M0 ∈ (0, m(t)̄ ], a finite time tf ∈ ?Γ , and a control τ(⋅) : [0, tf ]→ ?(τ) such that along the controlled trajectory (x(t), m(t)) = Γ(t, τ, x̃(τ), m̃ (τ)) on [0, tf ], we have x(t) ∈ ? on [0, tf ], x(tf) = xf, and m(tf) > M0.

One cannot have controllability for OIP for every value of τmax > 0. Indeed, pick a point (xi , mi) in For every control τ(⋅) taking values in ?(τmax), the corresponding controlled trajectory (x(⋅), m(⋅)) = Γ(⋅, τ, xi , mi) converges to Γ(⋅, 0, xi , mi) on [0, tp(xi)] as τmax tends to zero. Then, since rp(xi) < rc, there exists t ∈ [0, tp(xi)] such that ‖ r(t) ‖< rc implying that (x(⋅), m(⋅)) = Γ(⋅, τ, xi , mi) is not admissible for every control τ(⋅) taking values in ?(τmax). For large values of τmax, we can steer (xi , mi) ∈ to (xf , mf) ∈ as described in the following lemma.

Lemma 3.12. For every β > 0, μ > 0, and point yi = (xi , mi) in there exists τmax such that the following holds:

Proof. In the light of the remark preceding the lemma, it is enough to find a value of τ̄ and a control τ(⋅) taking values in ?(τ̄) steering yi = (xi , mi) to some point in along an admissible trajectory. We proceed as follows. Let C0 = ‖ri‖1/2‖υi‖which belongs to (0, Choose now the function υ(⋅) defined on some time interval [0, T] ̄ (with T ̄ to be fixed later) as follows: υ(0) =i, ṙ(t) = υ(t), and

Here r(t)⊥ denotes a continuous choice of vector perpendicular to r(t) in the 2D planespanned by ri and υi. (Implicitly, we assume with no loss of generality that υi ≠ 0 with ai > 0 and bi ≠ 0.) For simplicity, we assume next that ai = bi = 1. The curve (r(⋅), υ(⋅)) defined previously can be explicitely integrated using polar coordinates for r(⋅) = r(⋅) exp(iθ(⋅)). One gets that, on [0, T],

After integration, one has for t ∈ [0, T],

One also checks that with eibeing a constant vector of unit normcollinear to ri × υi and One deduces that r0p2(t) = C1r(t)1/2 for some positive constant C1. One thus fixes T so that rp(T̄) > rc. It remains to determine τ̄ so that (r(t), υ(t)) is part of a controlled admissible trajectory of the system Σε. One first integrates over [0, T] the differential equation

and then takes for t ∈ [0, T]. The final bound τ̄ is simply the maximum of ‖τ(t)‖ over [0, T].

As a combination of Lemmas 3.11 and 3.12, we obtain the following result.

Corollary 3.13. For every β > 0, μ > 0, and initial point (xi , mi) ∈ there exists a limiting value τmax > 0 depending on (xi , mi) such that the following properties hold:

The limiting value τmax can be computed by combining a shooting method and a bisection method as described in Section 11.4.

11.3.3Controllability for DOP

Let us define a system Σ̃sat associated with Σsat as

where all variables are the same as defined in equation (2.1). For every and measurable control function τ taking values in ?(τmax) with τmax, we define by Γ̃(t, τ(t), yi) the corresponding trajectory of Σ̃sat for some positive times.

Remark 3.14. For every controlled trajectory (r(t), υ(t), m(t)) = Γ(t, τ(t), yi) of the system Σsat on some finite intervals [0, tf] ⊂ ?Γ with (rf , υf , mf) = Γ(tf , τ(tf ), yi), the trajectory (r̃(t), ῦ̃(t), m̃ (t)) = Γ̃(t, τ(tf − t), rf , −υf , mf ) of Σ̃sat runs backward in time along the trajectory Γ(t, τ(t), yi) on [0, tf ], that is,

As a consequence, according to Lemma 3.12 and Corollary 3.13,we obtain the following result.

Corollary 3.15. For each β > 0, μ > 0, and mi > 0, given a point (rf , υf) ∈ ?−, there exists a τmax > 0 depending on (rf , υf ) and mi such that the following properties hold:

11.4.1A numerical example for OIP

In order to be able to compute the limiting value τmax in Corollary 3.13, we first define the following optimal control problem (OCP).

Definition 4.1 (Optimal control problem (OCP) for OIP). Given every initial point (xi , mi) ∈ and τ > 0, the OCP for OIP consists of steering a satellite by τ(⋅) ∈ ?(τ) on a time interval [0, tf] ⊂ ?Γ along the system Σsat such that, along the controlled trajectory (r(t), υ(t), m(t)) = Γ(t, τ(t), xi , mi), the time tf is the first occurrence for ‖ r(tf) ‖= rp(r(tf ), υ(tf )), that is, ‖ r(t) ‖> rp(r(t), υ(t)) on [0, tf ), and that rp(r(tf ), υ(tf )) is maximized, that is, the cost functional is

Let tf̃ >0 be the optimal final time of the OCP for OIP and (x̃(t), m̃ (t)) = Γ(t, τ̃(t), xi , mi) on [0, tf̃ ] be the optimal controlled trajectory with the associated optimal control τ̃(t) ∈ ?(τ) on [0, tf̃ ]. One can check, by using the Pontryagin maximum principle as was done in [7], that ‖ τ̃(t) ‖= τ on the whole interval [0, tf̃ ]. Thus, fixing the initial point (xi , mi) ∈ , we have that the final time tf̃ , the trajectory (x̃(t), m̃ (t)) at each time t ∈ [0, tf̃ ], and the final perigee distance rp(x̃(tf̃ )) are functions of τ. Thus, let us define a function

If one can find τmax > 0 such that s(τmax) = 0, then τmax is the limiting value in Corollary 3.13. For every τ > 0, using a shooting method as an inner loop to solve the OCP for OIP, we can obtain a value for s(τ). Then, using a bisection method as an outer loop, one can obtain τmax > 0 such that s(τmax) = 0. According to equation (2.14) and the objective of the OCP for OIP, place a satellite with the initial mass mi > 0 on a point (ri , υi) ∈ ?−. The optimal controlled trajectory lies on a 2D plane spanned by ri and υi. Hence, the limiting value τmax in Corollary 3.13 is determined only by ‖ ri ‖, ‖ υi ‖, and by the flight path angle ηi ∈ [−π/2, π/2], that is, the angle between the velocity vector υi and the local horizontal plane, defined by

Assume that a rocket carries a satellite whose initial mass is mi = 150 kg, from the surface of the Earth to a point xi = (ri , υi) in the unstable region ?− such that ‖ ri ‖= re + 110,000 m, ‖ υi ‖= 7879.5 m/s, and ηi = 5∘. The rocket and the satellite are separated at this point xi. Then, the satellite has to use its own engine to steer itself from the point xi into the stable region ?+. We can see from Figure 4 that the periodic orbit γxi has collisions with the surface of the atmosphere around the Earth.

We choose the specific impulse of the engine fixed on the satellite as Isp= 2000 s, which implies that where g0 = 9.8 m2/s. The computed result of the limiting value is τmax = 8.052 N. Thus, in order to be able to insert the satellite from the point xi into the stable region ?+, the maximum thrust of the engine has to be larger than 8.052 N. The optimal controlled trajectory of the corresponding OCP for OIP with τ = 8.052 is plotted in Figure 4 as well. To see the numerical resultsfor different initial points, other two points x1 = (r1, υ1) and x2 = (r2, υ2) are chosen on the periodic orbit γxi such that

‖ r1 ‖ = re + 379,494 m, ‖ υ1 ‖= 7,562 m/s, η1 = 4.3517° ,

‖ r2 ‖ = re + 599,351 m, ‖ υ2 ‖= 7,312m/s, η2 = 3.0132° .

Then, the limiting value of τmax corresponding to the two initial points xj, j = 1, 2, are computed as 9.037 N and 10.719 N, respectively.We see that the limiting values τmax are different for different initial points on the same periodic orbit. The time history of radius ‖ r(t) ‖ and the perigee distance rp(x(t)) along the optimal controlled trajectories starting from the initial points xi and xj, j = 1, 2, are plotted in Figure 5. Since the three points xi and xj, j = 1, 2 lie on the same periodic orbit γxi , rp at the initial time is the same, as shown in Figure 5.

11.4.2A numerical example for DOP

A DOP is a powered flight phase of a satellite in the region , during which a decelerating manoeuvre is performed so that the satellite will move to the desired final point xf = (rf , υf) ∈ ?− at the entry interface (EI). The condition at EI permits thesatellite to have a subsequent safe entry flight in atmosphere to a landing site.Atypical condition at EI, see [1], is given as

where rEI = re + 122,000 m, VEI = 7879.5m/s, and ηEI = −15° denote the norm of position vector, the norm of velocity vector, and the flight path angle at EI, respectively. In order to compute the limiting value τmax in Corollary 3.15 for the DOP to a point (rf , υf ) in ?−, we first define the below OCP.

Definition 4.2 (OCP for DOP). Given every final point xf = (rf , υf) ∈ ?− and τ > 0, let mi > 0 be the initial mass of a satellite, the OCP for DOP consists of steering the satellite by τ(⋅) ∈ ?(τ) on a time interval [0, tf ] ⊂ ?Γ subject to the system Σ̃sat such that, along the controlled trajectory (r(t), υ(t), m(t)) = Γ̃(t, τ(t), rf , −υf , mf ) (mf > 0 is free) of the System Σ̃sat, the time tf is the first occurrence for ‖ r(tf ) ‖= rp(r(tf ), υ(tf )), mi = m(tf ), and rp(r(tf ), υ(tf )) is maximized, that is, the cost functional is the same as equation (4.1).

Given every initial mass mi > 0 and final point (rf , υf ) in ?−, let tf̄ > 0 be the optimal final time of the OCP for DOP and (x̄(t), m̄ (t)) = Γ̃(t, τ̄(t), rf , −υf , mf ) be the optimal controlled trajectory associated with the control τ̄(t) ∈ ?(τ) on [0, tf̄ ]. Then, the same as the OCP for OIP, the perigee distance rp(x̄(tf̄ )) is a function of τ. Let us define a function

Then, according to equation (3.3), in order to compute the limiting value τmax in Corollary 3.15, it suffices to combine a shooting method and a bisection method to compute the value τmax such that s̄(τmax) = 0.

What we developed in this paper is applicable not only for low-thrust control systems but also for high-thrust control systems if only the thrust is finite instead of impulsive. Thus, we consider the space shuttle’s parameters in [3, 10]. The initial mass is 95,254.38 kg. The specific impulse of the engine is 313 s, which means β = 3.26 × 10−4. The numerical result is τmax = 14,004.62N. Note that the propulsion for a space shuttle is provided by the orbital manoeuvring system (OMS) engines which produce a total vacuum thrust of 53,378.6 N; see [3, 10]. Thus, according to Lemma 3.11, for every initial point xi in ?+, the space shuttle can reach the EI condition in equation (4.3) by admissible controlled trajectories of the system Σsat if the satellite takes enough fuel. The periodic trajectory γxf and the associated optimal controlled trajectory with τ = τmax are illustrated in Figure 6. The profile of ‖ r ‖ and rp along optimal controlled trajectories for the DOP with τ = τmax, τmax + 100 N, and τmax −100 N, are illustrated in Figure 7.We can see from Figure 7 that the optimal controlled trajectory of the OCP for DOP with τ = τmax +100N is an admissible controlled trajectory in ? and the final point lies in ?+. While, the optimal controlled trajectory of the OCP for DOP with τ = τmax − 100N cannot reach a point in ?+ by admissible controlled trajectories of the system Σ̃sat.